Fake Tori, the Annulus Conjecture, and the Conjectures of Kirby

The main result of this note (Theorem A) is that the set of piecewise linear (P.L.) manifolds of the same homotopy type as the n-torus, Tn, n ≥ 5, is in one-to-one correspondence with the orbits of Λ n-3(π 1Tn) ⊗ Z2 under the natural action of the automorphism group of π 1Tn. Every homotopy torus has a finite cover P.L. homeomorphic to Tn; hence the generalized annulus conjecture holds in dimension ≥ 5 (Kirby, R. C., ``Stable homeomorphisms,'' manuscript in preparation). The methods of this classification are also used to study some conjectures of R. C. Kirby (manuscript in preparation) related to triangulating manifolds.